F79MA: Statistical Modelling - Estimation Mean Square Error Results - Statistics Assessment Answer

Statistics Assessment Task

Project description You are a statistical trainee actuary working for an insurance company. You are part of a statistical modelling team that is considering using the negative binomial distribution to model the number of claims arising from a certain portfolio of policies. The probability mass function of a negative binomial random variable X is given by P(X = x|θ) = Γ(r + x) Γ(r)Γ(x + 1)θ r (1 − θ) x , x = 0, 1, 2 . . . with parameters θ ∈ [0, 1] and r > 0. Note that E(X|θ) = r(1 − θ)/θ.

As a member of this team, you have been asked to explore the statistical properties of the maximum likelihood estimator for θ, assuming that r is fixed. (Note: assume r = 5). You decide to perform the following analyses:

1. Derive the maximum likelihood estimator for θ (denoted henceforth by ˆθ).

2. Use simulation in R to characterise the bias of ˆθ as a function of the size of the random sample n. Check that the bias vanishes as n → ∞ for θ ∈ (0, 1) and briefly discuss your results.

3. Use simulation in R to characterise the variance of ˆθ as n increases (use n = 1, . . . , 50). Plot the estimation mean square error as a function on n and for θ ∈ (0, 1). Briefly discuss your results. 

4. Derive the Fisher information for θ and use it to calculate the Cramer-Rao bound for an unbiased estimator for θ. 

5. Compare the estimation mean square error results found in part 3 with the theoretical bound of part 4 and briefly discuss your results.

6. Would the method of moments estimator for θ produce better or worse estimation results in terms of mean square error? Explain your answer.

This Statistics Assessment has been solved by our Statistics experts at My Uni Paper. Our Assignment Writing Experts are efficient to provide a fresh solution to this question. We are serving more than 10000+ Students in Australia, UK & US by helping them to score HD in their academics. Our Experts are well trained to follow all marking rubrics & referencing style.

Be it a used or new solution, the quality of the work submitted by our assignment experts remains unhampered. You may continue to expect the same or even better quality with the used and new assignment solution files respectively. There’s one thing to be noticed that you could choose one between the two and acquire an HD either way. You could choose a new assignment solution file to get yourself an exclusive, plagiarism (with free Turnitin file), expert quality assignment or order an old solution file that was considered worthy of the highest distinction.